On Riesz subsets of Abelian discrete groups
Let G be a compact abelian group, and \(\Gamma\) its dual. A subset \(\Lambda\) of \(\Gamma\) is a Riesz set if every bounded measure whose Fourier coefficients vanish off \(\Lambda\) is absolutely continuous with respect to the Haar measure of G. The theorem of F. and M. Riesz (1916) is that the set of positive integers is a Riesz subset of the integers. The paper here is an encyclopedic collection of results on Riesz sets, some old, some new. For example, it includes a proof of the Mooney-Havin theorem that the quotient \(L^ 1(T)/H^ 1(D)\) is weakly sequentially complete. Let \(\sigma\) be the Haar measure of G. Then f is in weak \(L^ 1\) if the products \(t\sigma (| f| >t)\), \(t>0\), are bounded. A subspace of \(L^ 1\) is nicely placed if its unit ball is closed in weak \(L^ 1\), while a subset \(\Lambda\) of the dual \(\Gamma\) is nicely placed if the space \(L^ 1_{\Lambda}\) of functions in \(L^ 1(G)\) whose Fourier coefficients vanish off \(\Lambda\) is nicely placed. Finally, \(\Lambda\) is a Shapiro set if every subset of \(\Lambda\) is nicely placed. Joel Shapiro (1978) proved that the set of positive integers is a Shapiro subset of the integers. The principal result of the paper here is that if \(\Lambda\) is a Shapiro set, then \(\Lambda\) is a Riesz set, and the quotient \(L^ 1/L^ 1_{\Lambda}\) is weakly sequentially complete. The paper includes several striking arithmetic results. For example, the set of squares of integers is closed in the Bohr compactification of the integers.
scientific article; zbMATH DE number 4080120
scientific article